Compute the intersection form

Question

Consider a smooth subvariety $\iota:X=V_+(f)\subset\mathbb{P}^2\times\mathbb{P}^1$ with some $f\in H^0(\mathcal{O}(1,2))$, how can I compute the intersection $$\iota^*\mathcal{O}(2,3).\iota^*\mathcal{O}(2,3)$$

Answer 1

It is convenient to write a divisor corresponding to $O(m,n)$ as $mH+nL$. Then, what you want is $(H+2L)\cdot (2H+3L)\cdot (2H+3L)$. Using distributivity, one only need to know $H^i\cdot L^{3-i}$. One checks that this is zero if $i\neq 2$ and $H^2\cdot L=1$. I will let you do the rest.